Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [825,4,Mod(1,825)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(825, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 0, 0]))
N = Newforms(chi, 4, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("825.1");
S:= CuspForms(chi, 4);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 825 = 3 \cdot 5^{2} \cdot 11 \) |
| Weight: | \( k \) | \(=\) | \( 4 \) |
| Character orbit: | \([\chi]\) | \(=\) | 825.a (trivial) |
Newform invariants
comment: select newform
sage: f = N[0] # Warning: the index may be different
gp: f = lf[1] \\ Warning: the index may be different
| Self dual: | yes |
| Analytic conductor: | \(48.6765757547\) |
| Analytic rank: | \(0\) |
| Dimension: | \(7\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{7} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{7} - 3x^{6} - 44x^{5} + 118x^{4} + 515x^{3} - 1279x^{2} - 892x + 1840 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 2^{2} \) |
| Twist minimal: | no (minimal twist has level 165) |
| Fricke sign: | \(1\) |
| Sato-Tate group: | $\mathrm{SU}(2)$ |
$q$-expansion
comment: q-expansion
sage: f.q_expansion() # note that sage often uses an isomorphic number field
gp: mfcoefs(f, 20)
Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{6}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.
Basis of coefficient ring in terms of a root \(\nu\) of
\( x^{7} - 3x^{6} - 44x^{5} + 118x^{4} + 515x^{3} - 1279x^{2} - 892x + 1840 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - 14 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( \nu^{6} - 2\nu^{5} - 14\nu^{4} - 24\nu^{3} - 261\nu^{2} + 1036\nu + 1200 ) / 80 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( \nu^{5} - 4\nu^{4} - 26\nu^{3} + 88\nu^{2} + 93\nu - 200 ) / 10 \)
|
| \(\beta_{5}\) | \(=\) |
\( \nu^{3} - 2\nu^{2} - 21\nu + 27 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( -3\nu^{6} + 6\nu^{5} + 122\nu^{4} - 168\nu^{3} - 1297\nu^{2} + 1132\nu + 2240 ) / 80 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + 14 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{5} + 2\beta_{2} + 21\beta _1 + 1 \)
|
| \(\nu^{4}\) | \(=\) |
\( \beta_{6} + 3\beta_{5} + 3\beta_{3} + 32\beta_{2} + 10\beta _1 + 294 \)
|
| \(\nu^{5}\) | \(=\) |
\( 4\beta_{6} + 38\beta_{5} + 10\beta_{4} + 12\beta_{3} + 92\beta_{2} + 493\beta _1 + 170 \)
|
| \(\nu^{6}\) | \(=\) |
\( 22\beta_{6} + 142\beta_{5} + 20\beta_{4} + 146\beta_{3} + 941\beta_{2} + 594\beta _1 + 6934 \)
|
Embeddings
For each embedding \(\iota_m\) of the coefficient field, the values \(\iota_m(a_n)\) are shown below.
For more information on an embedded modular form you can click on its label.
comment: embeddings in the coefficient field
gp: mfembed(f)
| Label | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | |||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 1.1 |
|
−4.66025 | 3.00000 | 13.7179 | 0 | −13.9808 | 20.4072 | −26.6471 | 9.00000 | 0 | |||||||||||||||||||||||||||||||||||||||
| 1.2 | −4.31207 | 3.00000 | 10.5939 | 0 | −12.9362 | −9.06309 | −11.1853 | 9.00000 | 0 | ||||||||||||||||||||||||||||||||||||||||
| 1.3 | −1.35250 | 3.00000 | −6.17074 | 0 | −4.05750 | −5.74924 | 19.1659 | 9.00000 | 0 | ||||||||||||||||||||||||||||||||||||||||
| 1.4 | 1.16334 | 3.00000 | −6.64664 | 0 | 3.49002 | 19.4748 | −17.0390 | 9.00000 | 0 | ||||||||||||||||||||||||||||||||||||||||
| 1.5 | 2.62783 | 3.00000 | −1.09451 | 0 | 7.88349 | −24.0582 | −23.8988 | 9.00000 | 0 | ||||||||||||||||||||||||||||||||||||||||
| 1.6 | 4.00699 | 3.00000 | 8.05595 | 0 | 12.0210 | 26.2766 | 0.224208 | 9.00000 | 0 | ||||||||||||||||||||||||||||||||||||||||
| 1.7 | 5.52667 | 3.00000 | 22.5440 | 0 | 16.5800 | 22.7119 | 80.3801 | 9.00000 | 0 | ||||||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(3\) | \(-1\) |
| \(5\) | \(-1\) |
| \(11\) | \(1\) |
Inner twists
This newform does not admit any (nontrivial) inner twists.
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 825.4.a.bc | 7 | |
| 3.b | odd | 2 | 1 | 2475.4.a.bq | 7 | ||
| 5.b | even | 2 | 1 | 825.4.a.bb | 7 | ||
| 5.c | odd | 4 | 2 | 165.4.c.a | ✓ | 14 | |
| 15.d | odd | 2 | 1 | 2475.4.a.br | 7 | ||
| 15.e | even | 4 | 2 | 495.4.c.c | 14 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 165.4.c.a | ✓ | 14 | 5.c | odd | 4 | 2 | |
| 495.4.c.c | 14 | 15.e | even | 4 | 2 | ||
| 825.4.a.bb | 7 | 5.b | even | 2 | 1 | ||
| 825.4.a.bc | 7 | 1.a | even | 1 | 1 | trivial | |
| 2475.4.a.bq | 7 | 3.b | odd | 2 | 1 | ||
| 2475.4.a.br | 7 | 15.d | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(825))\):
|
\( T_{2}^{7} - 3T_{2}^{6} - 44T_{2}^{5} + 118T_{2}^{4} + 515T_{2}^{3} - 1279T_{2}^{2} - 892T_{2} + 1840 \)
|
|
\( T_{7}^{7} - 50T_{7}^{6} - 98T_{7}^{5} + 36272T_{7}^{4} - 352032T_{7}^{3} - 4759608T_{7}^{2} + 42636816T_{7} + 297322848 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{7} - 3 T^{6} + \cdots + 1840 \)
|
| $3$ |
\( (T - 3)^{7} \)
|
| $5$ |
\( T^{7} \)
|
| $7$ |
\( T^{7} - 50 T^{6} + \cdots + 297322848 \)
|
| $11$ |
\( (T + 11)^{7} \)
|
| $13$ |
\( T^{7} + \cdots - 9421357536 \)
|
| $17$ |
\( T^{7} + \cdots + 367433036800 \)
|
| $19$ |
\( T^{7} + \cdots - 300905291776 \)
|
| $23$ |
\( T^{7} + \cdots + 545164650496 \)
|
| $29$ |
\( T^{7} + \cdots - 95816988486144 \)
|
| $31$ |
\( T^{7} + \cdots - 75419306700800 \)
|
| $37$ |
\( T^{7} + \cdots + 419226771456 \)
|
| $41$ |
\( T^{7} + \cdots - 15\!\cdots\!56 \)
|
| $43$ |
\( T^{7} + \cdots - 6383171396736 \)
|
| $47$ |
\( T^{7} + \cdots - 1789387955200 \)
|
| $53$ |
\( T^{7} + \cdots - 14\!\cdots\!08 \)
|
| $59$ |
\( T^{7} + \cdots - 13\!\cdots\!60 \)
|
| $61$ |
\( T^{7} + \cdots + 45\!\cdots\!20 \)
|
| $67$ |
\( T^{7} + \cdots - 41\!\cdots\!00 \)
|
| $71$ |
\( T^{7} + \cdots + 20\!\cdots\!84 \)
|
| $73$ |
\( T^{7} + \cdots + 13\!\cdots\!72 \)
|
| $79$ |
\( T^{7} + \cdots - 60\!\cdots\!68 \)
|
| $83$ |
\( T^{7} + \cdots + 89\!\cdots\!00 \)
|
| $89$ |
\( T^{7} + \cdots - 66\!\cdots\!08 \)
|
| $97$ |
\( T^{7} + \cdots - 15\!\cdots\!00 \)
|
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